Calculating numbers of servers for tiers of a multi-tiered system

ABSTRACT

Fractional, non-integer numbers of servers are calculated for respective tiers of a multi-tiered system using a server allocation algorithm. The fractional, non-integer numbers of servers are rounded up to integer numbers to compute allocated numbers of servers for respective tiers.

CROSS-REFERENCE TO RELATED APPLICATIONS

This is a continuation-in-part of U.S. Ser. No. 10/706,401, filed Nov. 12, 2003, which is incorporated by reference herein.

This is a continuation-in-part of U.S. Ser. No. 11/116,728, entitled “Allocation of Resources for Tiers of a Multi-Tiered System Based On Selecting Items From Respective Sets,” and of U.S. Ser. No. 11/117,197, entitled “Allocating Resources in a System Having Multiple Tiers,” both filed on Apr. 28, 2005, and both hereby incorporated by reference.

BACKGROUND

Web-based resources, such as online information, online retail sites, and other web resources, are widely available to users over networks such as the Internet and intranets. Access to a web resource is provided by a website, which is typically implemented with a server system having one or more servers. Traffic at a popular website can be relatively heavy. If an insufficient number of servers are allocated to a website, then response times experienced by users when visiting the website can be long.

Typically, a server system that provides a website has a multi-tiered architecture that includes multiple tiers of servers. A user request submitted to the server system is typically processed by all the tiers. Thus, the total response time for a user request is the sum of the response times at respective tiers. The expected response time at each tier depends upon the number of servers allocated to the tier.

A web-based, multi-tiered architecture typically includes three tiers of servers: web servers, application servers, and database servers. Web servers communicate with the user (at a client system) and serve as the interface between the application servers and the user. Application servers parse user requests and perform predefined computations. Application servers in turn communicate with database servers to access requested information.

Conventional techniques have been proposed to allocate an adequate number of servers to each tier of a multi-tiered architecture to meet a desired response time constraint. However, conventional techniques for allocating the number of servers in each tier of a multi-tiered architecture employ algorithms that have processing times related exponentially to the input problem size. In other words, as the number of tiers of the multi-tiered server system increases, the processing time for computing the allocation of servers in each tier increases exponentially. Consequently, the amount of processing time involved in performing server allocation using conventional techniques for a multi-tiered server system can be relatively large. Also, conventional techniques are inflexible in that, in a particular system, users typically are limited to using one server allocation algorithm to address the server allocation issue.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of server systems and a server management system according to an embodiment of the invention.

FIG. 2 is a flow diagram of a process performed by a server allocation module executable in the server management system, in accordance with an embodiment.

DETAILED DESCRIPTION

FIG. 1 illustrates components of an example multi-tiered server system 100A, which includes multiple tiers of servers. The server system 100A includes a first tier of web servers 102, a second tier of application servers 104, and a third tier of database server(s) 106. In one example, the server system 100A is used for web-based applications (e.g., e-commerce, search engine, web service, and so forth). However, in other embodiments, the multi-tiered server system can include other types of servers for other applications, such as storage applications, utility computing shared environment applications, and so forth. Also, even though three tiers are depicted for the server system 100A in FIG. 1, it is contemplated that other multi-tiered server systems can include other numbers of tiers, such as two or greater than three.

FIG. 1 also shows another multi-tiered server system 10B, which can be configured in similar manner as server system 100A, or can be configured differently with different types of servers and different numbers of tiers. The example arrangement of FIG. 1 also shows a server pool 108, which includes a number of spare servers that are dynamically allocated to either server system 100A or server system 100B to expand the capacity of the server system 100A or 100B based on demands experienced by either server system 100A or 100B. In the embodiment of FIG. 1, the server pool 108 includes spare web servers 110, spare application servers 112, and spare database servers 114. One or more of each of the spare web servers 110, spare application servers 112, and spare database servers 114 are dynamically added to corresponding tiers of either the server system 100A or server system 100B to dynamically expand capacity at each respective tier.

For example, as user traffic at a server system increases, servers from the server pool 108 are added to one or more tiers of the server system to assure that average user response time is below some predefined threshold. In some cases, the predefined threshold is specified by a service level agreement (SLA) entered between the provider of the server system and another party, such as the party providing the e-commerce service, search engine service, or web service, as examples.

A “server” refers to a computer, a collection of computers, or any part of a computer (whether software or hardware or a combination of both).

As depicted in FIG. 1, client systems 116 are able to access the web servers 102 to issue requests (e.g., user requests) to the server system 100A. The same client systems 116, or other client systems, can issue requests to the server system 100B.

A server management system 118 (implemented as a computer in some embodiments) is coupled to the server systems 100A, 100B, and server pool 108. The server management system 118 includes a server allocation module 120, which in one example implementation is a software module (or plural software modules) executable on one or plural central processing units (CPUs) 122 in the server management system. The CPU(s) 122 is (are) coupled to storage 124.

The server management system 118 also includes a server monitor 126 that is executable on the CPU(s) 122. The server monitor 126 is used to monitor the status of each server system 100A, 100B. The server monitor 126 receives data (e.g., amount of user traffic, etc.) from each server system 100A, 100B and stores the data in the storage 124 for use by the server allocation module 120. Note that the tasks performed by the server monitor 126 and by the server allocation module 120 can be combined such that a single software module performs the tasks. Alternatively, the tasks can be performed by a larger number of software modules.

Also, although described in the context of software modules, it is contemplated that the server allocation and server monitoring tasks can be performed by hardware, or by a combination of hardware or firmware, in other embodiments.

The server allocation module 120 is able to dynamically allocate a number of servers to each tier of a multi-tiered server system (100A or 100B) depending upon the incoming workload (requests from client systems). The server allocation module 120 computes a number of servers allocated to each tier such that the total server system cost is reduced while a desired average response time is achieved for requests from the client systems 116. Average response time is one example of a response time constraint. Another example of a response time constraint includes maximum response time. Based on information collected by the server monitor 126, additional resources (in the form of servers) are dynamically allocated by the server allocation module 120 from the server pool 108 to the server system 100A or 100B to add resources to each tier to achieve the desired average response time. In another embodiment, the server allocation module 120 is able to allocate numbers of servers to respective tiers of a multi-tiered server system without using data from the server monitor 126.

Although reference is made to allocating servers to tiers of a server system, it is contemplated that other types of resources (e.g., CPUs, memory, etc.) can be allocated to plural tiers of any multi-tiered computer, according to alternative embodiments.

In accordance with some embodiments of the invention, for enhanced flexibility, the server allocation module 120 selects one of multiple server allocation algorithms to use for determining a system configuration (that includes a number of servers in each tier of the multi-tiered server system) such that total system cost is minimized while a response time constraint is satisfied.

In one embodiment, the server allocation module 120 is able to selectively use one of three algorithms, including a first server allocation algorithm 128, a second server allocation algorithm 130, and a third server allocation algorithm 132. In other embodiments, a smaller number or larger number of server allocation algorithms can be selectively used.

The first server allocation algorithm 128 (referred to as the “two-approximation algorithm”) is relatively fast since the first server allocation algorithm 128 computes an approximate solution with a total system cost that is at most twice the optimal cost. More generally, the first server allocation algorithm 128 computes an approximate solution with a total system cost that is at most m times the optimal cost, where m is greater than one. The term “optimal cost” refers to the minimum cost that can be achieved by an optimum configuration of the multi-tiered server system while still satisfying the response time constraint.

The second server allocation algorithm 130 (referred to as the “pseudo-polynomial-time exact algorithm”) computes an optimal (or exact) solution, rather than an approximate solution, in pseudo-polynomial running time (as contrasted with the exponential running time associated with some conventional techniques for server allocation). The term “pseudo-polynomial running time” refers to the polynomial running time of an algorithm if all input numbers have bounded sizes, where the algorithm typically runs in exponential or other longer running time otherwise.

The third server allocation algorithm 132 (referred to as the “fully-polynomial-time approximation algorithm”) computes an approximate solution with the total cost at most (1+ε) times the optimal cost, where the running time is a polynomial in both 1/ε and k, where k is the number of tiers.

Note that the first and third algorithms produce feasible solutions regarding the numbers of servers for respective tiers that satisfy the response time constraint, but the solutions are not optimal (but rather are approximate) in the sense that the solutions do not achieve optimal (e.g., minimum) cost. However, the second algorithm produces a solution that achieves optimal (e.g., minimum) cost. Note also that the first algorithm is usually faster than the third algorithm, since the value of ε is a constant between 0 and 1.

A property of a multi-tiered server system architecture is that the delay experienced by a request in each tier depends upon the number of servers in that tier and is not affected by the number of servers in any other tier. Thus, according to some embodiments, the response time for a request can be computed by summing the delays in individual tiers to obtain the total response time.

In one embodiment, the server allocation module 120 of the server management system 118 uses a response time model that assumes that all servers in the same tier are identical, and that the workload is shared approximately equally among all the servers in the same tier. The server allocation module 120 performs server resource allocation modeled as an integer optimization problem, with one linear objective function and an arbitrary decomposable additive constraint, as follows:

$\begin{matrix} {{\min\;{\sum\limits_{i = 1}^{k}\;{h_{i}\; N_{i}}}}{{s.t.\;{\sum\limits_{i = 1}^{k}\;{f_{i}\;\left( N_{i} \right)}}} \leq T}} & \left( {{Eq}.\mspace{14mu} 1} \right) \end{matrix}$

N₁, . . . , N_(k) are positive integers.

In the equation, h_(i) represents a cost of a server, and ƒ_(i)(N_(i)) represents a function for calculating a response time based on a number N_(i) of servers at tier i. Conventionally, allocating servers to different tiers of a multi-tiered computing system with the objective of minimizing system cost while meeting a response time constraint across all the tiers typically is NP-hard in general (where computation time is exponential). However, according to some embodiments of the invention, polynomial-time approximation algorithms (first and third server allocation algorithms 128 and 132) and a pseudo-polynomial-time exact algorithm (130) are selectively used to allocate server resources. The approximation algorithms run in polynomial time and offer the flexibility to control the tradeoff between solution quality and running time. In other words, approximate server allocation solutions can be computed to quickly obtain results in certain situations. If more accurate results are desired, then the exact algorithm 130 is used. The ability to select one of plural server allocation algorithms provides enhanced flexibility for changing system environments.

Use of the approximation algorithms (e.g., first and third server allocation algorithms 128 and 132) provide smaller computation times than exponential-time based server allocation algorithms. Also, as described further below, the pseudo-polynomial-time algorithm 130 is transformed into a multi-choice knapsack problem that simplifies the computation of numbers of servers in a multi-tiered server system.

Using server allocation techniques according to some embodiments, user control of running time and solution quality is provided. Selection (either by user or the server management system 118) of the approximation algorithms (first or third algorithms) results in reduced computation time and reduced solution quality, but selection of the optimal algorithm (second algorithm) results in increased computation time, but increased solution quality. As discussed further below, control of usage of storage space can also be accomplished by selecting a server allocation algorithm with reduced storage usage.

In the response time model according to an example implementation, if the request arrival rate is λ_(i) (incoming workload) for the i-th tier, and this tier has N_(i) servers, then each server has a request arrival rate of λ_(i)/N_(i). In one implementation, each server is modeled as a processor-sharing queue. The expected response time is given by the expected time in a system with an M/G/1 queuing model (Poisson distribution for request interarrival density, arbitrary distribution for service processing time, and one single queue with unbounded buffer length):

$\begin{matrix} {{{R_{i}\;\left( N_{i} \right)} = \frac{E\left\lbrack S_{i} \right\rbrack}{1 - {\left( \frac{\lambda_{i}}{N_{i}} \right)\;{E\left\lbrack S_{i} \right\rbrack}}}},} & \left( {{Eq}.\mspace{14mu} 2} \right) \end{matrix}$ where R_(i)(N_(i)) represents the average response time of the i-th tier, S_(i) is a random variable denoting the processing time (or service demand) of a request on a resource (such as a CPU) at the i-th tier, and E[S_(i)] is the expected processing time of a request on a resource. In other implementations, other queuing models can be employed.

The parameter E[S_(i)] can be estimated from the measured utilization rate, u_(i), of the resource as follows: E[S_(i)]=u_(i)/(λ_(i)/N_(i)). The average response time, R(N), for a k-tier server system is the sum of the delays in all the tiers. Therefore,

$\begin{matrix} {{{R\;(N)} = {{\sum\limits_{i = 1}^{k}\;{R_{i}\;\left( N_{i} \right)}} = {\sum\limits_{i = 1}^{k}\frac{E\left\lbrack S_{i} \right\rbrack}{1 - \frac{\lambda_{i}\;{E\left\lbrack S_{i} \right\rbrack}}{N_{i}}}}}},} & \left( {{Eq}.\mspace{14mu} 3} \right) \end{matrix}$ where N=(N₁, N₂, . . . , N_(k)). The parameter N is referred to as a configuration, and represents the numbers of servers in respective tiers.

From Eq. 3, it follows that there exist multiple server configurations satisfying a given response time constraint. Among these multiple feasible allocations, it is desired to find the one configuration that has the minimum cost. Finding the configuration with the minimum cost can be formulated as the following optimization problem:

$\begin{matrix} {\min\limits_{N_{i}}\;{\sum\limits_{i = 1}^{k}\;{h_{i}\; N_{i}}}} & \left( {{Eq}.\mspace{14mu} 4} \right) \\ {{{{s.t.\mspace{14mu} R}\;(N)} = {{\sum\limits_{i = 1}^{k}\;\frac{E\left\lbrack S_{i} \right\rbrack}{1 - \frac{\lambda_{i}\;{E\left\lbrack S_{i} \right\rbrack}}{N_{1}}}} \leq T_{0}}};} & \left( {{Eq}.\mspace{14mu} 5} \right) \end{matrix}$

N_(i) integer with N_(i)>λ_(i)E[S_(i)], for i=1, . . . , k,

where T₀ is the response time threshold value and the weights h_(i) (all assumed to be positive) are the costs of each server in the different tiers. In other words, h_(i) represents the cost of a server in the i-th tier. In the case where costs are difficult to ascertain, a convenient simplification involves minimizing the total number of servers Σ_(i=1) ^(k)N_(i); that is, to set h_(i)=1 for all i. Because:

${\frac{E\left\lbrack S_{i} \right\rbrack}{1 - \frac{\lambda_{i}\;{E\left\lbrack S_{i} \right\rbrack}}{N_{i}}} = {\frac{N_{i}\;{E\left\lbrack S_{i} \right\rbrack}}{N_{i} - {\lambda_{i}\;{E\left\lbrack S_{i} \right\rbrack}}} = {{E\left\lbrack S_{i} \right\rbrack} + \frac{\lambda_{i}\;{E\left\lbrack S_{i} \right\rbrack}^{2}}{N_{i} - {\lambda_{i}\;{E\left\lbrack S_{i} \right\rbrack}}}}}},$ for all i, the non-linear constraint (Eq. 5) can be further simplified as follows:

${\sum\limits_{i = 1}^{k}\;\frac{\lambda_{i}\;{E\left\lbrack S_{i} \right\rbrack}^{2}}{N_{i} - {\lambda_{i}\;{E\left\lbrack S_{i} \right\rbrack}}}} \leq {T_{0} - {\sum\limits_{i = 1}^{k}\;{{E\left\lbrack S_{i} \right\rbrack}.}}}$ Let a_(i)=λ_(i)E[S_(i)]², b_(i)=λ_(i)E[S_(i)] and T=T₀−Σ_(i=1) ^(k)E[S_(i)], then the response time constraint (in this example the average response time constraint) becomes:

$\begin{matrix} {{\sum\limits_{i = 1}^{k}\;\frac{a_{i}}{N_{i} - b_{i}}} \leq {T.}} & \left( {{Eq}.\mspace{14mu} 6} \right) \end{matrix}$

Note that the response time constraint takes into account the incoming workload, λ_(i), from each tier i. Even though server allocation is generally NP-hard for a variable number of tiers, real-world systems are usually composed of a relatively small number of tiers. For example, a typical e-commerce application has three tiers. Where the number of tiers is constant, it is possible to solve the server allocation problem in polynomial time or pseudo-polynomial time, using algorithms according to some embodiments.

FIG. 2 shows a server allocation process performed by the server allocation module 120 (FIG. 1), according to an embodiment. In response to receiving (at 202) a request (such as through the user interface 134, through the network interface 136, or from elsewhere) to perform server allocation, the server allocation module 120 determines (at 204) which algorithm to select to perform the server allocation. The received request includes a request from a user, or a request automatically initiated in the server management system 118 or from another source.

Selection of the algorithm (at 204) is based on various factors, including user input, as well as factors relating to how accurate the results should be. For example, a user may expressly indicate which of the three algorithms to use for performing the server allocation process. Alternatively, the server allocation module 120 can determine, from the context of the request, whether the desired server allocation determination is to be highly accurate or just a rough approximation. In one example context, a user may be accessing the server management system 118 (FIG. 1) “on-line” (such as the user interactively using the server management system 118 to obtain answers regarding server allocation in the server system 100A or 100B (FIG. 1) to meet incoming workloads). In such a context, use of either the first or third approximation algorithms would likely provide a quicker answer (approximate answer) than the second algorithm. Note that the first algorithm (the two-approximation algorithm) usually also has a faster running time than the third approximation algorithm (depending on the value set for ε in the third approximation algorithm).

Thus, in the context where a relatively quick answer is desirable, the server management system 118 (FIG. 1) selects one of the approximation algorithms. In a different context, where computation time is not an issue and a more accurate solution is desirable, the server management system selects the second server allocation algorithm. More generally, the server management system determines, based on one or more conditions, which of the server allocation algorithms to select. The one or more conditions include a specific indication in the request issued by the user (user selection), or other conditions relating to the context in which a request for server allocation was submitted (e.g., user is on-line and would like an answer quickly) which provides some indication of the desired computation time and solution quality.

In response to determining that the first algorithm 128 (FIG. 1), which is the two-approximation algorithm, is to be used, the server allocation module 120 relaxes (at 206) the constraint that each N_(i) (the number of servers at the i-th tier) should be an integer. According to the two-approximation algorithm, the Lagrangian multiplier method is used to compute a closed-form solution for the relaxed optimization problem (where the value of each N_(i) does not have to be integer). The relaxed optimization problem is also referred to as a continuous-relaxation model. According to this algorithm, the output value obtained for each N_(i) is a fractional (or continuous) value. To be meaningful, the fractional value is rounded up to the nearest integer. Therefore, according to this method, the approximate solution provides a solution with a cost that is less than twice the minimum cost, and at most one extra server per tier.

The Lagrangian multiplier method uses a Lagrangian function where λ is the Lagrangian multiplier:

$\begin{matrix} {{L\;\left( {N_{1},N_{2},\ldots\mspace{14mu},N_{k},\lambda} \right)} = {{\sum\limits_{i = 1}^{k}\;{h_{i}\; N_{i}}} + {\lambda\;\left( {{\sum\limits_{i = 1}^{k}\;\frac{a_{i}}{N_{i} - b_{i}}} - T} \right)}}} & \left( {{Eq}.\mspace{14mu} 7} \right) \end{matrix}$

Note that N₁ represents the number of servers at tier 1, N₂ represents the number of servers at tier 2, and N_(k) represents the number of servers at tier k. As discussed above, the expression

$\sum\limits_{i = 1}^{k}\;{h_{i}\; N_{i}}$ represents the total cost of having N_(i) servers at tier i for i=1, . . . k (in a k-tier system). The expression

$\left( {{\sum\limits_{i = 1}^{k}\;\frac{a_{i}}{N_{i} - b_{i}}} - T} \right)$ represents the response time constraint. The following equation represents the partial derivative of the expression of Eq. 7 with respect to N_(i), which is set to zero:

$\begin{matrix} {{\frac{\partial L}{\partial N_{i}} = {{h_{i} - {\lambda \cdot \frac{a_{i}}{\left( {N_{i} - b_{i}} \right)^{2}}}} = 0}},{i = 1},\ldots\mspace{14mu},{k.}} & \left( {{Eq}.\mspace{14mu} 8} \right) \end{matrix}$

Solving for N_(i) in Eq. 8, an optimal fractional value N_(i) ^(ƒ) is obtained (at 208) as follows:

$\begin{matrix} {{N_{i}^{f} = {b_{i} + \sqrt{\frac{\lambda\; a_{i}}{h_{i}}}}},{i = 1},\ldots\mspace{14mu},{k.}} & \left( {{Eq}.\mspace{14mu} 9} \right) \end{matrix}$

The fractional value N_(i) ^(ƒ) represents the optimal number (a fractional value) of servers at tier i. Since the N_(i) ^(ƒ) values are continuous, the non-linear constraint is binding:

$\begin{matrix} {{\sum\limits_{i = 1}^{k}\;\frac{a_{i}}{N_{i}^{f} - b_{i}}} = {T.}} & \left( {{Eq}.\mspace{14mu} 10} \right) \end{matrix}$

Substituting the expression set equal to N_(i) ^(ƒ) in Eq. 9 into the equality of Eq. 10 and simplifying algebraically, the server allocation module 120 solves (at 210) for the Lagrangian function multiplier λ as follows:

$\begin{matrix} {\sqrt{\lambda} = {\frac{\sum\limits_{i = 1}^{k}\;\sqrt{h_{i}\; a_{i}}}{T}.}} & \left( {{Eq}.\mspace{14mu} 11} \right) \end{matrix}$

Then the cost of the relaxed Lagrangian solution is calculated (at 212) as follows:

$\begin{matrix} {{{cost}\;\left( N^{f} \right)} = {{\sum\limits_{i = 1}^{k}\;{h_{i}\; N_{i}^{f}}} = {{\sum\limits_{i = 1}^{k}\;{h_{i}\; b_{i}}} + {\frac{\left( {\sum\limits_{i = 1}^{k}\;\sqrt{h_{i}\; a_{i}}} \right)^{2}}{T}.}}}} & \left( {{Eq}.\mspace{14mu} 12} \right) \end{matrix}$

The optimal fractional solution N_(i) ^(ƒ) is converted into a feasible two-approximation solution directly by rounding up (at 214) the optimal fractional solution N_(i) ^(ƒ) to an approximate integer solution N_(i) ^(r). Specifically, N_(i) ^(r)=|N_(i) ^(ƒ)|, for all i. Because N_(i) ^(r)≧N_(i) ^(ƒ) for all i, Σ_(i=1) ^(k)a_(i)/(N_(i) ^(r)−b_(i))≦Σ_(i=1) ^(k)a_(i)(N_(i) ^(ƒ)−b_(i))=T. Therefore, the rounded up integer values N_(i) ^(r) for i=1, . . . , k, satisfy the response time constraint and N_(i) ^(r) is a feasible, approximate solution to the server allocation problem. Furthermore, N_(i) ^(r) gives a two-approximation to the server allocation problem, in other words, cost (N^(r))≦2·cost(N*), where N* is an optimal solution. The value cost (N^(r)) is the cost associated with N_(i) ^(r) servers in the k tiers (i=1, . . . , k). Usually, the value of cost(N^(r)) is greater than cost(N*).

An output of the two-approximation server allocation algorithm (128 in FIG. 1) includes the values N_(i) ^(r), i=1, . . . , k, which are approximate numbers of servers at respective tiers i. The cost, cost (N^(r)), associated with implementing these numbers of servers at the tiers i is at most twice the optimal cost, or 2·cost(N*). The computation time for computing the two-approximation server allocation algorithm is linear time (related linearly to the number of tiers).

Note that the two-approximation algorithm is applicable to other response time functions as well.

If the server allocation module 120 determines (at 204) that the algorithm to select is the pseudo-polynomial-time exact algorithm (second server allocation algorithm 130 in FIG. 1), then the server allocation module defines (at 216) k sets of items S₁, . . . , S_(k) for k tiers. The sets S₁, . . . , S_(k) are sets used for a multi-choice knapsack problem. The server allocation problem according to the second server allocation algorithm 130 becomes an algorithm that solves for the multi-choice knapsack problem optimally. The second server allocation algorithm 130 is a pseudo-polynomial-time algorithm that runs in polynomial time if all input numbers are bounded above by a constant. Such algorithms are technically exponential but display polynomial-time behavior in many practical applications. The NP-hard multi-choice knapsack problem is pseudo-polynomial-time solvable using the second algorithm 130 according to some embodiments.

The multi-choice knapsack problem is a generalization of the ordinary knapsack problem, in which there are m sets of items S₁, . . . , S_(m) with |S_(i)|=n_(i) for all i and Σ_(i=1) ^(m)n_(i)=n. Each item jεS_(i) includes a weight w_(ij) and a profit p_(ij), and the knapsack has capacity W. The objective is to pick exactly one item from each set, such that the profit sum is maximized and the weight sum is bounded by W.

When applied to the server allocation problem with k tiers, the multi-choice knapsack problem includes k sets of items S₁, . . . , S_(k). For an item jεS_(i), parameters weight(j) and cost(j) are calculated (at 218) as follows:

$\begin{matrix} {{{{weight}\;(j)} = \frac{a_{i}}{j - b_{i}}},{{{cost}\;(j)} = {j \cdot {h_{i}.}}}} & \left( {{Eq}.\mspace{14mu} 13} \right) \end{matrix}$

The parameter weights) represents the response time for j servers in tier i, and the parameter cost(j) represents the cost of j servers at tier i. For the server allocation optimization problem, there is no cost constraint; thus any j>b_(i) is a feasible item in S_(i). Thus, each set S_(i) has plural items j, representing numbers of servers, with each item j associated with parameters weight(j) (response time for j servers) and cost(j) (cost of j servers).

To reduce the size of input sets S_(i), and thus achieve polynomial-time computation time, the number of servers allocated at each tier is bounded within a lower bound n_(i) ^(l) and an upper bound n_(i) ^(u), calculated at (220). Based on the response time constraint, the lower bound n_(i) ^(l) is defined as follows:

$\begin{matrix} \left. {\frac{a_{i}}{N_{i} - b_{i}} \leq {\sum\limits_{i = 1}^{k}\;\frac{a_{i}}{N_{i} - b_{i}}} \leq T}\Rightarrow{N_{i} \geq {b_{i} + \frac{a_{i}}{T}} \equiv {n_{i}^{t}.}} \right. & \left( {{Eq}.\mspace{14mu} 14} \right) \end{matrix}$

Thus, n_(i) ^(l) is a lower bound on the number of servers to be allocated at the i-th tier, where the lower bound n_(i) ^(l) is calculated based on the response time constraint (see Eq. 6 above). Next, to calculate the upper bound, the two-approximation solution N_(i) ^(r) is considered (according to the first server allocation algorithm). Let C^(r)=cost(N^(r)). Since C^(r) is an upper bound on the total cost of the optimal solution, n_(i) ^(u) is defined as follows:

$\begin{matrix} \left. {{{h_{i}\; N_{i}} + {\sum\limits_{j|{j \neq i}}^{\;}\;{h_{j}\; n_{j}^{l}}}} \leq {\sum\limits_{j = 1}^{k}\;{h_{j}\; n_{j}}} \leq C^{r}}\Rightarrow{N_{i} \leq \frac{C^{r} - {\sum\limits_{j|{j \neq i}}^{\;}\;{h_{j}\; n_{j}^{l}}}}{h_{i}} \equiv {n_{i}^{u}.}} \right. & \left( {{Eq}.\mspace{14mu} 15} \right) \end{matrix}$

Thus, n_(i) ^(u) is an upper bound on the number of servers to be allocated at the i-th tier, where n_(i) ^(u) is calculated based on the cost associated with the two-approximation algorithm.

Given n_(i) ^(l) and n_(i) ^(u), the server allocation module 120 (FIG. 1) has to just consider items jεS_(i) satisfying ┌n_(i) ^(l)┐≦j≦└n_(i) ^(u)┘. Without loss of generality, it is assumed that n_(i) ^(l) and n_(i) ^(u) are both integers. Otherwise n_(i) ^(l) can be replaced by ┌n_(i) ^(l)┐ and n_(i) ^(u) can be replaced by └n_(i) ^(u)┘. Then set S_(i) has n_(i) ^(u)−n_(i) ^(l)+1 items, which reduces the number of items involved in the calculation to solve for N=(N₁, N₂, . . . , N_(k)), the optimal configuration.

Solving the multi-choice knapsack problem causes selection of items from respective sets S₁, . . . , S_(k) that satisfy the response time constraint and a target cost (e.g., an optimal cost). In one example, the optimal cost is the minimum cost. The multi-choice knapsack problem can be solved in pseudo-polynomial time by building (at 222) a dynamic programming (DP) table. There are two ways to build the dynamic programming table, using either the cost or the weight. As the weights of items are not integers, and the costs are integers, the DP table is efficiently built using the cost dimension. The DP table is a two-dimensional table having k rows (corresponding to the k tiers) and C^(r) columns (where C^(r) is the cost associated with the two-approximation algorithm described above). Each cell of the DP table (which contains k×C^(r) cells) has a value F(i,c), which represents the minimum weight (or response time) of items selected from the first i item sets with total cost bounded by c, for i=1, . . . , k, and c=1, . . . , C^(r). The i-th row of the DP table represents the optimal solution, minimum response time F(i,c), for the first i tiers. The first row of the DP table, F(l,c) is relatively easy to compute: F(l,c)=min {weight(j)|j·h₁≦c}, c=1, . . . , C^(r). Each field (representing the response time) of the first row F(l,c) is the minimum response time weight(j), such that j·h₁≦c.

Once the first row is computed, the subsequent rows (2, 3, . . . , k) of the DP table F(,) are built using the following recursion function:

$\begin{matrix} {{F\;\left( {i,c} \right)} = {\min\limits_{j \in S_{i}}{\left\{ {{F\;\left( {{i - 1},{c - {{cost}\;(j)}}} \right)} + {{weight}\;(j)}} \right\}.}}} & \left( {{Eq}.\mspace{14mu} 16} \right) \end{matrix}$

According to Eq. 16, F(i,c) is an aggregate value derived by summing the F(,) value from a previous row F(i−1, c−cost(j)) and the weight value, weight(j) (response time), of the current row, based on a selected j (number of servers) from set S_(i). Set S_(i) has multiple j values (representing different numbers of servers), in the range n_(i) ^(l)≦j≦n_(i) ^(u).

F(i,c) represents the minimum response time given a fixed cost c (derived based on cost(j)) for the first i tiers. From the two-approximation algorithm, C^(r) (the cost of N^(r)) is within a factor of two of cost(N*), the optimum cost. The size of the cost dimension (number of columns) in the DP table is C^(r). As a result, the total time involved in building the table is O(C^(r)··Σ_(i=1) ^(k)(n_(i) ^(u)−n_(i) ^(l)))=O(cost(N^(ƒ))·Σ_(i=1) ^(k)(n_(i) ^(u)−n_(i) ^(l))).

Once the DP table has been built, an optimal solution is found (at 224) by going through the last row, F(k,c), of the DP table and choosing the minimum cost c, such that F(k,c) is bounded by T (which is the response time constraint).

In other words, in step 224, the server allocation module 120 walks through the last row F(k) and chooses the minimum c value (c₀) such that F(k, c₀)≦T. The value c₀ is the minimum cost of the server allocation problem (aggregated for all k tiers) with the total response time, F(k,c₀), bounded by T.

Let A be an output array to hold the optimal solution, where each entry of the output array A is equal to the optimal number of servers for a respective tier. In other words, A[i], i=1, . . . , k, represents the optimal number of servers in the i-th tier. To calculate the values for A, the server allocation module 120 performs (at 226) a backtrack algorithm to find the optimal number of servers at each tier i.

The backtrack algorithm starts with i=k, c=c₀, and continues through a loop until i=0 (i=k, . . . , 0). The backtrack algorithm walks through the i-th row and finds a value jεS_(i), such that F(i,c)=F(i−1, c−cost(j))+weight(j). Effectively, in the i-th row, the algorithm identifies the entry with the minimum response time. That entry of the i-th row corresponds with a j value (that represents the number of servers at the i-th tier). Then the value A[i] is set as A[i]=j, and the values of i and c are updated as i←i−l, c←c−cost(j). Effectively, for row k of the DP table, A[k]=j, where j is the value where F(k,c₀)=F(k−1, c₀−cost(j))+weight(j).

In terms of performance, the above algorithm runs in time O(C^(r)·Σ_(i=1) ^(k)(n_(i) ^(u)−n_(i) ^(l)))=O(cost(N^(ƒ))·Σ_(i=1) ^(k)(n_(i) ^(u)−n_(i) ^(l))). The second server allocation algorithm thus runs in pseudo-polynomial time, which is less than an exponential time.

As explained above, the two-approximation algorithm (the first server allocation algorithm 128 in FIG. 1) provides an approximated solution with a cost that is less than twice the minimum cost. On the other hand, the pseudo-polynomial-time exact algorithm (the second server allocation algorithm 130 in FIG. 1) provides an exact solution that runs in pseudo-polynomial time. In accordance with some embodiments of the invention, an intermediate solution also can be achieved by using the third server allocation algorithm 132 (FIG. 1), which is a fully-polynomial-time approximation scheme. The approximate solution provided by the third server allocation algorithm 132 has better accuracy than the two-approximation algorithm.

The pseudo-polynomial-time algorithm described above can be converted into a fully-polynomial-time approximation scheme using cost scaling. The approximation scheme is almost identical to the pseudo-polynomial time algorithm, with the difference that a scaled down cost for each item is used and calculated (at 228) as follows:

$\begin{matrix} {{{{scost}\;(j)} = \left\lceil \frac{{k \cdot {cost}}\;(j)}{ɛ \cdot C^{r}} \right\rceil},{\forall{j \in {S_{i}.}}}} & \left( {{Eq}.\mspace{14mu} 17} \right) \end{matrix}$

The scaled down cost scost(j) replaces the cost(j) value associated with the second server allocation algorithm that is associated with items j in each set of the multi-choice knapsack problem. The value ε is selected based on the level of accuracy desired—a low ε value provides greater accuracy but involves greater processing time. Let S_(i)′ denote the i-th set with items associated with scaled costs. In other words, each set S_(i)′ includes items j that are each associated with scaled costs rather than the actual costs. If two items in the set have the same scaled cost, then the one with less weight (response time) is kept while the other item is discarded. The scost values are integer values calculated by rounding down the computed value according to Eq. 17. By using the scaled down cost values, the number of items in each set S_(i)′ of the multi-choice knapsack problem is further reduced for greater computational efficiency. However, the solution provided is an approximate solution, whose accuracy is based on the value of ε.

The set S_(i)′ is computed (at 230), where S_(i)′ contains k·1/ε different scost values. Each set S_(i)′ thus contains at most k/ε different items. S_(i)′ is computed efficiently according to the following algorithm. For each fixed t from 1 to k/ε, consider all the items jεS_(i) such that scost(j)=t. The maximum value of j, denoted j_(t), can be calculated in O(1) time (or constant time). The weight (response time) of the maximum j_(t) value becomes weight(j_(t))=a_(i)/j_(t)−b_(i)). The item with scost=t and weight=weight(j_(t)) is placed into set S_(i)′. The procedure to compute S_(i)′ takes both time and space O(k·1/ε).

To save storage space, not all the S_(i)′ have to be computed in advance. Based on the generated sets S_(i)′, a DP table is built (at 232), using a technique similar to that used for the pseudo-polynomial-time exact algorithm, described above. For improved efficiency, when computing the DP table, the whole DP table does not have to be stored. Instead, the entries F(i,c) of the DP table can be computed row by row. For i=1, . . . , k, F(i,c) is computed for all possible values of c. At the time of computing the i-th row F(i), first S_(i)′ is computed, then the row F(i) is computed using the values of the previous row F(i−1) and S_(i)′. After the i-th row F(i) is finished, S_(i)′ and F(i−1) are discarded, and the server allocation module 120 continues by processing the (i+1)-th row. As a result, the total storage usage for this algorithm is O(k·1/ε). Note that a similar storage savings technique can be applied to the second server allocation algorithm described above.

With costs scaled, the cost of the optimal solution is bounded by k·1/ε. There are k rows of the DP table, with each row of the DP table having length k/ε (that is, B=k/ε, where B represents the number of columns in the DP table), and each cell takes time O(k/ε) to compute. Thus, the total processing time becomes O(k³·1/ε²) In summary, this third server allocation algorithm involves a processing time O(k³·1/ε²) and storage space O(k·1/ε).

Each item of a set S_(i)′ with scaled cost induces an error bounded by (ε·C^(r)/k). There are a total of k items in the solution; thus, the total error in the cost is bounded by ε·C^(r). Since C^(r) is a two-approximation of the optimal cost, it follows that the total error is bounded by ε·C^(r)≦2ε·cost(N*). By replacing ε by ε/2, the time and space complexities of the above algorithm will change only by a constant factor, thus asymptotically the same bounds hold. This gives a (1+ε)-approximation to the server allocation problem. Using the third server allocation algorithm, the number of servers N_(i)′ computed for each tier i is a feasible allocation of the number of servers for that tier-however, this feasible allocation of the number of servers does not achieve an optimal cost, but rather, a (1+ε)-approximation of the optimal cost. In other words, the cost of the approximate solution N_(i)′ for i=1, . . . , k satisfies cost(N′)≦(1+ε)·cost(N*).

Instructions of such software routines described above are loaded for execution on processors (such as CPU(s) 122 in FIG. 1). The processors include microprocessors, microcontrollers, processor modules or subsystems (including one or more microprocessors or microcontrollers), or other control or computing devices. As used here, a “controller” refers to hardware, software, or a combination thereof. A “controller” can refer to a single component or to plural components (whether software or hardware).

Data and instructions (of the software) are stored in respective storage devices, which are implemented as one or more machine-readable storage media. The storage media include different forms of memory including semiconductor memory devices such as dynamic or static random access memories (DRAMs or SRAMs), erasable and programmable read-only memories (EPROMs), electrically erasable and programmable read-only memories (EEPROMs) and flash memories; magnetic disks such as fixed, floppy and removable disks; other magnetic media including tape; and optical media such as compact disks (CDs) or digital video disks (DVDs).

In the foregoing description, numerous details are set forth to provide an understanding of the present invention. However, it will be understood by those skilled in the art that the present invention may be practiced without these details. While the invention has been disclosed with respect to a limited number of embodiments, those skilled in the art will appreciate numerous modifications and variations therefrom. It is intended that the appended claims cover such modifications and variations as fall within the true spirit and scope of the invention. 

1. A method comprising: calculating, by at least one processor, fractional, non-integer numbers of servers for respective tiers of a multi-tiered system using a server allocation algorithm; and computing, by the at least one processor, allocated numbers of servers for respective tiers by rounding up the fractional, non-integer numbers to integer numbers, wherein calculating the fractional, non-integer numbers of servers for respective tiers of the multi-tiered system using the server allocation algorithm comprises using a Lagrangian multiplier technique.
 2. The method of claim 1, wherein the allocated numbers of servers for respective tiers are approximate numbers, and wherein computing the allocated numbers of servers for respective tiers of the multi-tiered system provides an approximate total system cost that is less than twice an optimal total system cost associated with an optimum solution for numbers of servers for respective tiers.
 3. The method of claim 2, wherein the optimum solution is defined as $\min\;{\sum\limits_{i = 1}^{k}\;{h_{i}\; N_{i}}}$ ${s.t.\mspace{14mu}{\sum\limits_{i = 1}^{k}\;{f_{i}\;\left( N_{i} \right)}}} \leq T$ N₁, . . . , N_(k) are positive integers, where h_(i) represents a cost of a server, k represents a number of tiers in the multi-tiered system, and ƒ_(i)(N_(i)) represents a function for calculating a response time based on a number N_(i) of servers at tier i, i=1 to k.
 4. The method of claim 1, wherein the Lagrangian multiplier technique employs a Lagrangian function, ${{L\;\left( {N_{1},N_{2},\ldots\mspace{14mu},N_{k},\lambda} \right)} = {{\sum\limits_{i = 1}^{k}\;{h_{i}\; N_{i}}} + {\lambda\;\left( {{\sum\limits_{i = 1}^{k}\;\frac{a_{i}}{N_{i} - b_{i}}} - T} \right)}}},$ where k represents the number of tiers in the multi-tiered system, λ is a Lagrangian multiplier, N₁ represents a number of servers at tier 1, N₂ represents a number of servers at tier 2, N_(k) represents a number of servers at tier k, h_(i) represents cost per server at tier i, and $\;\left( {{\sum\limits_{i = 1}^{k}\;\frac{a_{i}}{N_{i} - b_{i}}} - T} \right)$ represents a response time constraint.
 5. The method of claim 4, further comprising: taking the partial derivative of L with respect to N_(i); and solving for N_(i) to compute the fractional, non-integer numbers of servers at respective tiers i, i equals 1 to k.
 6. The method of claim 5, wherein the fractional, non-integer numbers are represented as N_(i) ^(ƒ), i equals 1 to k, and wherein calculating the fractional, non-integer numbers of servers for respective tiers i comprises solving ${N_{i}^{f} = {b_{i} + \sqrt{\frac{\lambda\; a_{i}}{h_{i}}}}},{i = 1},\ldots\mspace{14mu},{k.}$
 7. The method of claim 1, wherein calculating the fractional, non-integer numbers of servers for respective tiers is based on a response time constraint.
 8. The method of claim 1, wherein calculating the fractional, non-integer numbers of servers for respective tiers is based on a constraint including an average response time.
 9. The method of claim 1, wherein computing the allocated numbers of servers for respective tiers provides a feasible solution that satisfies a response time constraint.
 10. A machine-readable storage medium containing instructions that upon execution by at least one processor cause a system to: calculate fractional, non-integer numbers of servers for respective tiers of a multi-tiered system using a server allocation algorithm; and compute allocated numbers of servers for respective tiers by rounding up the fractional, non-integer numbers to integer numbers, wherein calculating the fractional, non-integer numbers of servers for respective tiers of the multi-tiered system using the server allocation algorithm comprises using a Lagrangian multiplier technique.
 11. The machine-readable storage medium of claim 10, wherein the allocated numbers of servers for respective tiers are approximate numbers, and wherein computing the allocated numbers of servers for respective tiers of the multi-tiered system provides an approximate total system cost that is less than twice an optimal total system cost associated with an optimum solution.
 12. The machine-readable storage medium of claim 10, wherein calculating the fractional, non-integer numbers of servers for respective tiers is based on an average response time constraint.
 13. The machine-readable storage medium of claim 12, wherein computing the allocated numbers of servers for respective tiers provides an approximate total system cost that is greater than an optimal system cost associated with an optimal solution.
 14. A method comprising: computing, by at least one processor, numbers of servers for respective tiers of a multi-tiered system using a server allocation algorithm, wherein the computed numbers of servers are part of an approximate solution in which the server allocation algorithm calculates fractional, non-integer numbers of servers for the respective tiers, wherein the computed numbers of servers are rounded up from the fractional, non-integer numbers; computing, by the at least one processor, a first cost for the computed numbers of servers for respective tiers of the multi-tiered system, wherein the first cost is less than m times a cost associated with an optimal solution for the numbers of servers in the respective tiers of the multi-tiered system, wherein m is greater than 1; and computing, by the at least one processor, a second cost based on the fractional, non-integer numbers of servers for the respective tiers.
 15. The method of claim 14, wherein computing the numbers of servers for respective tiers of the multi-tiered system uses the server allocation algorithm in which a constraint that the numbers of servers for the respective tiers are integers is relaxed.
 16. The method of claim 14, wherein computing the numbers of servers for the respective tiers of the multi-tiered system uses the server allocation algorithm that includes a Lagrangian multiplier technique.
 17. The method of claim 16, wherein computing the numbers of servers for the respective tiers is performed by using the Lagrangian multiplier technique to compute a closed-form solution for a relaxed optimization problem in which a constraint that the numbers of servers for the respective tiers are integers is relaxed.
 18. The method of claim 14, wherein computing the first cost comprises computing the first cost that is less than twice the cost associated with the optimal solution.
 19. The method of claim 14, wherein the multi-tiered system has k tiers, and wherein the numbers of servers for respective tiers comprise N_(i) ^(r), i=1 to k, the method further comprising: calculating N_(i) ^(ƒ), i=1 to k, for the respective k tiers, wherein N_(i) ^(ƒ), i=1 to k, comprise the fractional, non-integer numbers, wherein computing N_(i) ^(r), i=1 to k, is based on rounding up N_(i) ^(ƒ), i=1 to k, to respective integer values.
 20. A system comprising: a processor; and a server allocation module executable on the processor to perform a server allocation algorithm to allocate numbers of servers in respective plural tiers of a multi-tiered server system, wherein the server allocation module computes the allocated numbers of servers in the respective plural tiers based on calculating fractional, non-integer numbers of servers for respective tiers, wherein the server allocation module calculates the fractional, non-integer numbers of servers for respective tiers by using a Lagrangian multiplier technique.
 21. The system of claim 20, wherein the server allocation module is executable to compute a total system cost associated with the allocated numbers of servers for the respective tiers, wherein the total system cost associated with the allocated numbers of servers is greater than a total system cost associated with an optimal solution for numbers of servers for the respective tiers.
 22. The system of claim 20, wherein the allocated numbers of servers is an approximate but feasible solution that satisfies a response time constraint.
 23. A system comprising: means for calculating fractional, non-integer numbers of servers for respective tiers of a multi-tiered system using a server allocation algorithm; and means for rounding up the fractional, non-integer numbers of servers for the respective tiers to derive integer numbers of servers for the respective tiers of the multi-tiered system, the integer numbers of servers representing an approximate, feasible solution that satisfies an average response time constraint, wherein the server allocation algorithm uses a Lagrangian multiplier technique.
 24. A system comprising: a processor; and a server allocation module executable on the processor to calculate fractional, non-integer numbers of servers for respective tiers of a multi-tiered system using a server allocation algorithm that includes a Lagrangian multiplier technique, wherein the Lagrangian multiplier technique employs a Lagrangian function, ${{L\;\left( {N_{1},N_{2},\ldots\mspace{14mu},N_{k},\lambda} \right)} = {{\sum\limits_{i = 1}^{k}\;{h_{i}\; N_{i}}} + {\lambda\;\left( {{\sum\limits_{i = 1}^{k}\;\frac{a_{i}}{N_{i} - b_{i}}} - T} \right)}}},$ where k represents the number of tiers in the multi-tiered system, k is a Lagrangian multiplier, N₁ represents a number of servers at tier 1, N₂ represents a number of servers at tier 2, N_(k) represents a number of servers at tier k, h_(i) represents cost per server at tier i, and $\left( {{\sum\limits_{i = 1}^{k}\;\frac{a_{i}}{N_{i} - b_{i}}} - T} \right)$ represents a response time constraint, the server allocation module to further take the partial derivative of L with respect to N_(i), and to solve for N_(i) to compute the fractional, non-integer numbers (N_(i) ^(ƒ)) of servers at respective tiers i, i equals 1 to k, the server allocation module to further round up the fractional, non-integer numbers N_(i) ^(ƒ) of servers for the respective tiers to derive integer numbers of servers for the respective tiers of the multi-tiered system, the integer numbers of servers representing an approximate, feasible solution that satisfies the response time constraint. 